Some Topologies Connected with Lebesgue Measure Séminaire De Probabilités (Strasbourg), Tome 5 (1971), P

SÉMINAIREDEPROBABILITÉS (STRASBOURG)

JOHN B.WALSH Some topologies connected with Lebesgue measure Séminaire de probabilités (Strasbourg), tome 5 (1971), p. 290-310

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ SOME TOPOLOGIES CONNECTED WITH LEBESGUE MEASURE

J. B. Walsh

One of the nettles flourishing in the nether regions of the field of continuous parameter processes is the quantity lim sup X. s-~ t When one most wants to use it, he can’t show it is measurable. A theorem of Doob asserts that one can always modify the process slightly so that it becomes in which case lim sup is the same separable, X~ s-t as its less relative lim where D is a countable prickly sup X, set. If, as sometimes happens, one is not free to change the process, the usual procedure is to use the countable limit anyway and hope that the process is continuous. Chung and Walsh [3] used the idea of an essential limit-that is a limit ignoring sets of Lebesgue measure and found that it enjoyed the pleasant measurability and separability properties of the countable limit in addition to being translation invariant. Doob has recently generalized and improved this, showing that there is a large class of topologies on the line enjoying similar properties, called T-topologies.(T for Lebesgue, of course:) The first three sections of the present article are an exposition of Doob’s work, including some remarks on progressive measurability,which wasn’t treated in the original. The last sections give some applications, first to potential theory and then to Markov chains. The two main properties of these topologies which are established here are first, that one can define separability relative to such a turns topology in a very natural way, and then every measurable process out to be separable, without benefit of standard modifications, and

= lim secondly, if X is a measurable process, then so is Y, where sup ’ Yt Xs’ s--~ t and in fact y usually has better measurability properties than X. 291

Let T be a topology on the line R. If A C R, then AT will denote the set of T-accumulation points of A. Lebesgue measure and Lebesgue outer measure will be denoted by m and m* respectively. We will be concerned with topologies T on R satisfying

(a) T is finer than the Euclidean topology; (b) a set of Lebesgue measure zero has no accumulation points;

Such a topology will be called a T-topology. If not otherwise stated, topological notions will refer to the usual topology on the extended line.

A for instance will mean the ordinary closure of A. For a real valued funct ion f on R we will def ine the T-cluster set of f at a point t to be , where N ranges over all deleted neighborhoods of t; we will denote this set by CT(f;t) .. We define also

fT(t) = f(s) = max and

fT(t) = f(8) = min

Then we have fT a.e. (m); for fT everywhere by, and each the have definition, for b ~ 0 ,, set {t: f ( t ) >~ fT( t )+b} can no accumulation points, and hence is of zero Lebesgue measure by (c). Thus fT a.e. and similarly a.e. There are many T-topologies. One of the most useful is the essential topology L. An L-neighborhood of t is just an ordinary neighborhood minus a set of Lebesgue measure zero. A point t is an accumulation point for a set A in this topology iff 0 for each neighborhood N of t. One can easily verify that

A is = = (i) L-open iff A 0 - Q ,, where 0 is open and m(Q) 0 ;;

’ (ii) L is the coarsest T-topology; (iii) the L-Borel field is exactly the Lebesgue measurable sets.

Here, (iii) is a direct consequence of (i). The essential topology has some interesting connections with the ordinary topology: (iv) if A C R then AL is closed and perfect in the ordinary topology. 292

To see this, let t be in the closure of AL. If N is a neighborhood of t there exists N is also an L-neighborhood of s so

. Thus tE AL. Further, by (c), so in particular N contains points of A other than t. It can be shown that this property characterizes L among T-topologies. A direct consequence of (iv) is that for a function f (v) f is upper semicontinuous in the ordinary topology; (vi) f is continuous iff f is L-continuous.

Here, (v) follows by writing {fL a} as the complement of which is closed If f is it is ~ , by (iv). continuous, and if f is then f = fL = clearly L-continuous, L-continuous, f ,, so f is both.lower and upper semicontinuous, hence continuous. Two other examples of T-topologies are afforded by the right and left L-topologies,respectively denoted Lr and Neighborhoods of t are of the form N~1 and respectively, where N is an L-neighborhood of t. Results quite similar to the above hold for these topologies. Quite a different example is given by the topology Td: a point t is a Td-limit point of a set A if lim 1/2b 0. It is easily verified that Td is in a fact T-topology. T. is called the approximate topology, and it is classical that a Lebesgue measurable function is approximately continuous almost everywhere.

2.. MEASURABLE PROCESSES

Let (Q,F,P) be a probability space. Let B denote the Borel sets and consider the product space RXQ. We will denote the completion

of BXF with respect to mXP by’ BXF. A function X on R 03A9 to R is Borel measurable if it is BXF measurable, and is Lebesgue measurable if it is BXF measurable. An additional property shared by some but not all of the T-topologies is 293

(d) If (t,w) is Borel measurable, then so

is (t,w) -~X~(t,u)) ..

Here as = usual T-lim sup X(s,w).. Note that (d) is not s2014~t satisfied by the ordinary topology ! We say that two functions X and

Y on RXQ are if a.e.. = indistinguishable for (P) w, X(t,w) Y(t,w), all t..

PROPOSITION 2.1 Let T be a T-topology which also satisfies (d) and let X be a Lebesgue measurable process. Then X T is indistinguishable from a Borel measurable process.

PROOF. There exists a Borel measurable Y such that Y = X mXP a.e.

Fubini’s a.e. w By theorem, for m{t: X(t,o)) ~ Y(t,cu)}= 0. By (b), then, for all such = for all t. By (d) Y T is Borel measurable, qed

PROPOSITION 2.2 The topologies Land T along with their right and left hand topologies satisfy (d).

PROOF. Suppose first that X is the indicator function of some set in BXF. Notice that in that case XL(t,03C9) = lim and = lim flim sup f-a -~ These are both B F measurable by Fubini’s theorem. For general

measurable = BXF X, notice that for any real number band T L or T = T ::

which is I{XLb} = lim (I{Xb-1/n}), BXF measurable, qed. The proof for the respective right and left hand topologies is entirely similar and is left to the reader. qed

What one often needs in applications is progressive measurability rather than just measurability. Let be an (Ft)t~Rf increasing set of Borel sub fields of and let be the F t) i ass 0 f Bo reI sets of (-~,a). Then a function X on R 03A9 is said to be progressive measurable if for each a~R the restriction X~ of X to (here it is important to take the open interval) is measurable 294

and is progressively Lebesgue measurable if Xa is B XF measurable, the completion being again with respect to m X P .. (We are ignoring for the moment the question of whether or not X is adapted to i.e. whether or not X(t,.) is F. measurable for each fixed t.

LEMMA 2.3 If X is progressively Lebesgue measurable there exists a

progressively Borel measurable process X such that X = X mXP-a.e.

PROOF For each integer n ~1 and each integer k there exists a function Y on 03A9 which is Bk/2n Fk/2n measurable and is mXP-a.e. to X ~ . Define Y on RXQ equal n by

= on , and finally def ine X lim inf . Yn~ = Y~’~ Ck-1 ~2n, k~2n ~ X ~ , Y~ Now X plainly equals X mXP-a.e. To see that X is progressively

= n and Borel measurable, note that if a k~2n ,, then for each m ~ is measurable. Thus the same is true of j f: k,, Y. B XF 1m and hence of X. For a general a, just apply this remark to a sequence of dyadic rationals which increase to a. qed

THEOREM 2.4 Let T be a T-topology which satisfies (d). If X is progressively Borel measurable, so is XT ;; if X is progressively Lebesgue measurable, XT is indistinguishable from a progressively Borel measurable process. In the first case XT is adapted to (F ) and in the second it is indistinguishable from an adapted process.

PROOF If for all a, Xa is B X F measurable, then by (d), so is (Xa)T. But (Xa)T = (XT)a, for Xa is the restriction of X to the open interval so XT is also progressively Borel measurable. If X is only progressively Lebesgue measurable, it is equal mXP-a.e. to a progressively Borel measurable X. We have just shown that XT A is progressively Borel measurable, and the argument in the proof of Proposition 2.1 shows XT and XT are indistinguishable. 295

The statements on adaptability of XT are consequences of the general fact that any process progressively Borel measurable with respect to (Ft) is adapted to (Ft+). Indeed, for any b > a, X(a,.) = Xb(a,.), while is measurable by Fubini’s theorem. Xb(a,.) Eb" qed

The reason for insisting on the transition from progressively Lebesgue to progressively Borel is that the important stopping theorems-

of for a measurability X(S) stopping time S, for instance--valid for progressively Borel processes are false for progressively Lebesgue processes.

3. SEPARABILITY

Suppose now that {Xt’ is a stochastic process.

If will denote . D CR ,, XL t E D ~ . According to a classical theorem of Doob, there exists a countable set D and a standard modification X of X such that for a.e. w, X is in the cluster set of ~ at t for all t £ R. If X is progressively Lebesgue or Borel measurable, one can choose X to be progressively Lebesgue or Borel measurable respectively. Recalling our notation for cluster sets, separability for X implies

(3.1) = and = C0r (X;t) ’ where 0 and Or are the Euclidean and Euclidean right hand topologies respectively. We will take (3.1) as a model for a definition of T-separability. Let T be a T-topology.

DEFINITION A real valued stochastic process {Xt’ is T-separable (right T-separable) if there exists a countable set DCR, called a T-separability set (right T-separability set ) and KCQ with P(K) - 0 such that if (M ~ K then

(3.2) CT(X;t) (resp. CTr(X;t) ~ C0r(X|D;t) ) for all t. 296

X is strongly T-separable (strongly right T-separable) and D is a strong (right) T-separability set if there is equality in (3.2).

REMARKS 1. If X and X are Lebesgue measurable standard modifications of each other, then wp1 they have exactly the same T-cluster sets for any T-topology T. This follows since by Fubini’s theorem, w.p.1

= hence the set has no T-accumulation m(t: X./ X ) 0, by (b) X ~ X~ points. 2. A super-set of a separability set is a separability set but a super-set of a strong separability set may not be a strong separability set..

THEOREM 3.1 Let T be a T-topology and let X be a Lebesgue measurable process. Then X is T-separable and right T-separable. X is also strongly L-separable and strongly right L-separable.

PROOF We will prove separability only; the same proof works for right separability. Fix for the moment a set N C. R with meN) = 0. Let X be a Lebesgue measurable standard modification of X and let D be its separability set. Then w.p.1 we have for all t:

= = ’ where the first equality follows from remark 1, the second by separability and the last because, as D is countable, XL and |D are indistinguishable. Thus X is T-separable. Now we specialize to the essential topology. So far the set N has played no role, but now we choose it as follows. Let g be continuous on the compact interval For each w, m-a.e. Since L satisfies (d), both sides are Lebesgue measurable so by Fubini’s theorem there is an m-null set Ng~R such that if t ~ Ng then w.p.1. Choose ~g ~ dense in and let N . With this N, let D be the separability set above. As DnN by definition, for each s ~ D for all s in D g(X ) (g(X ))~ .. This is true w.p.1 simultaneously 297

so, noting that is upper semicontinuous by (v),

lim sup

The opposite inequality follows since D is an L separability set. Thus there is equality simultaneously for all t E R and g E C~-~,o~~ .. But this implies strong separability since if a is a limit point of XDD at t, choose to have a strict maximum 1 at a. Then 1 = lim sup s-~ t g(Xs) = s£D so a is in the L.cluster set of X at t. qed

We had no scruples about using the strongest form of Doob’s separability theorem above, even though it is decidedly deeper than the theorem in question. For a demonstration more or less by hand of a similar theorem the reader may glance at §5 of We should warn the reader that we have given a slightly stronger definition of T-separability than Doob. With his definition, all Lebesgue measurable processes are strongly separable relative to any T satisfying (d). With our definition, we have the curious fact that = this is true only for T L.. Indeed, if T is different from L there will be some Borel set AC R for which AL. As L is coarser than there T, exists t o in A~- A. Let f be the indicator function of A. Note that = = fT(to) 0 while fL(to) 1 . But f = f m-a.e. so is an accumulation by (c), to point of {f = 1~ ..

By T-separability, it is also an accumulation point of = Jf , where D is any T-separability set. Thus the T-cluster set of f at while the cluster set to is of ,at to also contains 1 . It follows that if Q is a probability space, the stochastic process = f(t) is T-separable but not strongly so. 298

4. APPLICATIONS TO POTENTIAL THEORY

One of the virtues of the limits we have been discussing is that they often exist where ordinary limits do not. For instance, nothing can be said about the existence of ordinary limits for a Lebesgue measurable submartingale t O J unless it is separable. On the other hand it has right and left L-limits (and therefore limits in any right or left T-topology) at all w.p.1. This is a simple consequence of L-separability and the existence of one-sided limits along any countable parameter set.

Another use of these limits is to sidestep thorny measurability problems. Let be a probability space and increasing family of Borel subfields of ~.. be a progressively Borel measurable stochastic process taking values in a

on locally compact metric space E. Define a complete measure 03BD E by by ~ (A) = E~ dt ~ for Borel sets A. ~ progressively Then, if f is 03BD-measurable, the process is/Lebesgue measurable: for there exist Borel measurable functions f’ and fit on E for which

f"j = 0 and f’&f&f"; then the set {(s,u): is both =t B XF =t measurable and of m)( P-measure null. It follows by (d)

That and are indistinguishable from progressively Borel measurable processes.

Now let us specialize this situation. Suppose, in the notation of [1], that X = is a Hunt process, except that we will not assume quasi left continuity of the paths. Measurability notions become more complex in this setting because of the presence of more than one measure on the probability space; in fact, for each probability measure~ on E there is a probability measure pM. on (Q,F). Recall that is the Borel field generated s t! and F0 =F0t. Then F and Ft are gotten by completing F0 and F0t respectively according to the prescription in [ lj. Notions of measurability in the following will be with respect to the fields F and Ft unless specifically stated otherwise. 299

We will say that a process Z is ( ro essivel Lebesgue measurable if for each probability measurer on E, Z is (progressively) Lebesgue measurable on the space The idea of progressive Borel measurability is independent of measure, but it is convenient to introduce another notion: a process Z is nearly (progressively) Borel measurable if for each probability measure on E there exists a (progressively) Borel measurable process Z such that Z and Z’~ are indistinguishable when the probability space is given the measure P~:, With these notions it is clear that the analogue of Theorem 2.4 is1

THEOREM 2.4’ Let T be a T-topology which sat isf ies (d). If Z is progressively Lebesgue measurable and real valued, then ZT is nearly progressively Borel measurable and adapted.

This theorem follows upon applying the remark of the previous paragraph separately for each P ~‘ The pleasant properties of progressive Borel measurability carry over to nearly progressively Borel measurability.

THEOREM 4.1 Suppose Z = fZt,t >,0 ~ is a nearly progressively Borel measurable process taking values in some separable metric space G, and S is a stopping time (always with respect to (Ft~.~ Then ZS is FS-measurable. Furthermore, if A is an analytic set in G and SA = inf ~t > 0: Zt E A~ , then SA is a stopping time.

PROOF Fix and let Z be a progressively Borel measurable process which is from Z. c P -indistinguishable If H F, H will designate the Borel field generated by g plus all the P~-null sets of

F .. By T49 of [ 5], is measurable and to Z~‘ Fg equalq Z P~-a.e..e. Thus is measurable Zs with respect to FS = F. On the other hand, according to Theorem T52 the random variable is a stopping time, equal to S A p"-a.e. Thus the set BS.? in hence in F ed 300

Hunt’s potential theory for strong Markov processes is based on the first hitting time of a set, but one can also base a potential theory on the idea of a first penetration time, as has been done for instance by Stroock L61. The first penetration time rrA of a set ACE is the infimum of times t for which A~ > 0, or, in our notation

(4.1) .

If A C E is universally measurable, or just R1-measurable for each probability measure on E, where R1(B) = J, then will be nearly progressively Borel measurable and rtA will be a stopping time. Notice that penetration times are measurable for a much larger class of sets than are first hitting times, and their measurability is a rather trivial fact. Nevertheless, we will see shortly that penetration times are in reality a subclass of hitting times.

It is natural to define a topology F, which we shall call the essentially fine topology, which is connected with penetration times. A set A is open in this topology if for each x E A there exists a universally measurable subset B CA such that and 01= 0 . The set of F-cluster points of A is denoted by AF. A closely related notion is that of essential regularity. A point x is essentially regular for A if for each universally measurable B C A, OJ= 0. In fact, the set A of essentially regular points coincides with AF if the resolvent charges no point, and in general we have

where the It is easy to see that AF =1 IBF and n B r sets B which intersections are taken over all universally measurable fine contain A. The closure of a set A in the essentially topology is given by AuA =AUA. 301

PROPOSITION 4.2 If A C E is Borel, so are AF and Ar. If A is universally measurable, both A and Ar are nearly Borel measurable.

PROOF Let g(t,w) = If A is Borel, this is measurable, hence so is Then = 1 E FO so 1} is Borel measurable. But A is exactly the set where this probability is one. If A is universally measurable and ~ is a probability measure on E, there exist Borel B and C such that BCACC and ( C-B) = 0. Then BrC ArC Cr and Br and Cr

’ are Borel measurable. X spends almost no time in C-B w.p.1 (P’~); we claim Cr- Br is polar for the process with initial distribution ~t. If not there is a stopping time S for which 0[5p. 1 ~,2 ~. But a.s. on {XS ~ Cr- BrJ we have 0 = implying the process spends strictly positive time in C - B, a contradiction. The proof for A is similar except one uses the function h(x,t,w) = instead of g. qed.

THEOREM 4.3 Let A be universally measurable. Then n = 03C0Ar = SAr w.p.1, and Ar a.s. on (where SB is the first hitting time of B.)

PROOF If S is a stopping time, w.p.1: iff = 1; for I A (XS) = 1 = 0 which is-by the strong Markov property - equivalent to == 0} = 1 . IA(Xt) is right upper semicontinuous in t, so = 1, implying a.e. on Applying the above remark to fixed times S and using Fubini’s theorem, we can show that the sets and {(s,w): = 1 ~

differ by a set of m x P measure zero. Thus w.p.1 , as both equal inf ~s: = 1 ~ .* Now consider SAr. By the section theorem 162~, for 6>0 we can find a stopping time S between SAr and SAr + E such that 0 a.e. on {XS ~ Ar} by the strong Markov property. Thus SAr w. p. 1 . This completes the proof since SAr is always less than or equal to 03C0Ar. qed . 302

COROLLARY Let A 6 E.. Then AF and Ar are closed in both the fine and essentially fine topologies.

Proof It is enough to show fine closure since the fine topology is coarser than the essentially fine topology. We need only consider universally measurable sets. Suppose x is regular for the nearly Borel set AF. If x we are done, so suppose it is not. Then there a time such that e 1 - 6. exists stopping S 6- ,, A ~ ~ > But A we have X S e as = 0 ~ = 1 ; no as = 1 or x E But A so _ ~=_~ , A . XS 6 =~ XS ~ x, by right of the paths > so in fact x E the desired continuity S xo 8 S 0; AF,’ contradiction. The proof for A r is similar. qed

This result gives some insight into why the penetration times can be defined on a larger class of sets than hitting times: penetration times reduce to hitting times of nearly Borel finely closed sets. We should note in passing that much of the above can be obtained from the general theory of additive functionals, for Ar is exactly the fine support of the continuous additive functional

. It is amusing, if not particularly significant, that the essentially fine topology satisfies a’) it is finer than the fine topology; b’) a set of potential zero has no accumulation points; c’) if A is universally measurable, A - A has potential zero.

Before leaving the subject we will give one more useful corollary.

THEOREM 4.4 Let f be a real valued universally measurable function which is continuous in the essentially fine topology. Then i) f is nearly Borel measurable; ii) f is fine continuous; iii) s -+ f(X ) is almost surely right continuous. 303

PROOF Write {x: f (x) >, a1 f(x) a - n-1}r. Each set in the intersection is nearly Borel and finely closed by the corollary. This proves i) and ii). The last part is now a well-known consequence, but we will give a short proof to illustrate the methods of this paper.

We may assume f is bounded. Set Z = . s f(X Then Z is upper semicontinuous from the right and, according to Theorem 2.4’, it is nearly progressively measurable. Thus T = infra: is a stopping time and ZT >. F .. The strong Markov property and essential fine continuity of f imply = = ~(X,p) ;; so is right continuous. It is therefore well measurable, as is and hence If some > T21 of [, X f(X s ). / f(X s )~ s~ 0 ,, by 5] there is a time S such that which stopping P ~ f(X ) ~ 0 ,, again contradicts the essential fine continuity of f, hence f(X) is identical to the right continuous process

Once it is known that ess.ential fine continuity (at all points!) implies fine continuity the above theorem can be derived trivially from known results. However, its novelty is precisely that only essentially fine continuity is needed. Observe, for instance, how quickly one can derive Hunt’s result that an excessive function is nearly Borel and right continuous along the sample paths. We need only check that an excessive function f is essentially finely continuous. Fix x e E.. Then ~f ( Xs )? s >. 0 ~ is a Lebesgue measurable supermartingale with values in [0,Co[ and thus has an essential limit f.. at s = 0 ;; t’his limit is constant with Px probability one by the Blumenthal zero-one law. But then, if s > 0

f(x) = Psf(Xs) .

Now let s--~ 0; then P f(x) -+ f(x), implying fo = f(x). Note that the above equation is valid even if some or all of the terms are infinite. 304

5. APPLICATIONS TO MARKOV CHAINS

Most of the interesting aspects--and pathologies--of the sample function behavior of continuous parameter Markov processes are already present in the relatively simple case of Markov chains. The sample functions can be everywhere discontinuous,for instance. One of the bothersome features of these processes is that in general there is no separable version of the process with values in the original space. One can compactify the space in some convenient manner and then find a separable version in the enlarged space, but then the process will spend some time (of Lebesgue measure zero) outside the original state space. This suggests the suitability of the topologies we have been discussing, for they have a tolerant astigmatism which allows them to overlook sets of measure zero. We will briefly discuss some of the sample function properties of a standard Markov chain from this viewpoint. Only the viewpoint has any claim to novelty; the theorems can all be found in sections II.4 - II.9 of

Let I = ~ O,i,2,... ~ and , be a standard transition matrix, that is, a transition matrix satisfying p..(t) ~ 03B4ij, as t -> 0 .. It is known that (5.1) pij(t) is continuous in t for t > 0, and q. - lim exists.

The state i is said to be if and unstab if = o~. stable , qi

Let t >. be a Markov chain transition matrix P. ~Xt,, 0 ~ having The continuity of the p..(t) imply that X is stochastically continuous and thus L 5] can be assumed to be Borel measurable.

PROPOSITION .1 Any countable dense set is a strong L, Lr, and L1 separability set..

PROOF If S and S’ are countable dense sets, and if s 6 S U S’, stochastic continuity implies that X is a.s. in the cluster sets of 305

both X IS and at s. Thus the cluster sets of and of are equal at all t w.p.1. If S is a strong separability set (and one exists by Theorem 3.1) then so is S~ The same remarks apply to the left and right cluster sets. qed

NOTE For f bounded on I we write 03A3pij(t)f(j) and R P f(i) Then for f>,O ,{Rpf(Xt), t 0} is a supermartingale and therefore has Lr and L1 limits at each point w.p.1. This immediately implies the following basic property of the paths.

PROPOSITION 5.2 If w is not is some set N with P(N) = 0, then t-+Xt(w) has at most one finite Lr and Ll cluster point at each t.

PROOF Let f = I. and note that, as the transition matrix is standard, pRpf as p --~oo . Given i ~ j, then, we can find p large enough so that R f(i) R p f(j) . Let Q - N be the set on which R p f(X) has Lr and L1 limits, and just notice that existence of an Lr limit (Ll limit) at t implies that not both i and j can be in the Lr clster set of X at t . qed

For i E I, define S . i (w ) _ ~ t : Xt (w) = i ~ . This set on the particular version of X chosen, but the sets dependsand SL1 are invariant under standard modification, up to a set of zero probability. Proposition 5.2 doesn’t exclude the possibility of having 4D as a cluster point at some, or even all t, but it does tell us that the sets S. 1 and S. J aren’t too badly mixed up. If i and j are distinct, then and are empty and 1 J 1 J S. L "S L is discrete.

PROPOSITION 5.3 W.p.1, Si, SLi, and SLri are identical up to sets of Lebesgue measure zero.

PROOF For any set A, = 0 and m(A - AL) = 0 . It remains to show m(SLri - S.) = 0. But, neglecting Lebesgue null sets: 306

SLri - Si = SLri[ Sj ] ~[ SLrj] which is empty by the remarks preceeding this proposition. qed

Notice that this proposition does not quite imply Theorem 3 p. lsl of [2$ which states in our notation that P (t m I . However, remark that m(S~ - Sll/d) m 0 -property (c)--so that by Fubini’s theorem this probability must be one for a.e. (m) t, and hence, as it is independent of t, for every t. As in §4, fl~ denotes the first penetration time of A, and we write 03C0i instead of

PROPOSITION Q/ For 0 £ qi m, and s, t z 0 ( 5.2) P ( s 6 Shr ) X = I ) = I ( 5.3) [ Xs = I) = e-qit.

PROOF (5.2) is a consequence of the remarks preceeding the proposi tion and th.e fact that L r is coarser than Tdr.. Suppose qi m.

( 5.4) P m I , k = °, .:., 2~ )I x~ = I) =

>= [l1 - 2 i + o(20(2 ~)]~~)

which converges to as This implies

(5.5) = I for all dyadic r~[0,1]|Xs = = I) = .

As any countable dense set is a strong L~ separability set this gives (5.3). Note that strong separability is needed for this implication. If q~ is infinite, (5.4) remains valid since for any N b I and large enough n the right hand side of (5.4) is dominated and hence

must be zero. qed 307

Sooner or later one usually wishes to make the process separable; this can be done in any compact separable Hausdorff space E containing I. The main advantage gained is commonly the availability of some form of the strong Markov property; with the right choice of E, one can actually find a strongly Markov standard modification of Xt. But even the simplest choice of E gives a useful form of it. Consider the process on the Alexandroff com- pactification {~} of I:

or in other language

= reals. ess lim inf X (ou) ,, taken in the extended s~t s By strong L r separability (the full force of strong separability is needed) this is indistinguishable from the lim inf over s~t Xjw)" rational s,.which is the version given by Chung. This process is Borel measurable, even lower semi continuous, and is a standard

modification of X by (5.2) and Proposition 5.2.. Any L separability set for S--which is to say any countable dense set--will be a

separability set for X+. Set . We already know all about this set because:

PROPOSITION 5.5 S. and SLr are indistinguishable.

PROOF Suppose w is in the set of p robability one in which SLri~ whenever . Then = (Xt)Lr i.. But if (Xt)Lr = j i, then we would have to have t E S.r, which can’t happen. Thus t 6 S~’ ~~ X. = i. The opposite implication J is obvious from the definitions of X and . qed

The process X+ is not strongly Markov for it may have oD for a value and there are no transition probabilities for this point. Never- theless X+ does satisfy a restricted version of the strong Markov property. Effectively, it is strongly Markov except when it is at ~D. Let Et denote the Borel field generated by s f t J . 308

PROPOSITION 5.6 Let T be a stopping time with the property that X- C I a.s. on T co. Then X satisfies the strong Markov property at T.

PROOF By remarks following the definition, is a cluster point of X~, s~T, s rational. But T CO«~ so in fact we see X is a limit from the right of rationals for which = X- . ~ Thus we can find a sequence T of rational valued stopping times such that

= Fix j 6 I and t ~ 0 . The rationals ~ X~}2014~P(Tc~. and the rationals translated by t are both separability sets hence ~T+t " ~ ~ ~’ Thus on ~ ~~ ’’

Markov conditioning on F we have by Fatou’s lemma and the property ~ lim sup I ~ .

By the choice of will be a limiting value of a.s. so this is ~ ~j0 - To see there is actually equality, note that a.s. on ~T ~J

1 = ~ ’~ which implies equality for all j. qed

Many results follow easily from this. For instance, for any time the t ~ is a stopping T, post-T process (.X-~ OJ again Markov chain with tha same transition probabilities. To see this,

to the times T + t t is chosen such apply proposition 5.6 n , where n that a. s. on Fubini’ s has this property) and let tn ~ o . We conclude with a final proposition which concludes our description of the sets SLri. 309

PROPOSITION 5.7 SLri is perfect in the right limit topology . if it is a.s. the union If ~ , it is a.s. nowhere dense; co, of left semi closed intervals, finite in each finite time interval.

PROOF For any set A C R, is perfect in the right limit topology.

SLri being closed in 0 ,, to show it is nowhere dense we need only show it contains no intervals. But it is BF measurable, so if it contained an interval with positive probability, Fubini’s theorem would tell us there is a t such that interior of . In S. P t E fact, as i and differ by sets of Lebesgue measure 0, i we can even require Si > 0}. But if qi = ~, 03C0I-{i} t -- 0 by Proposition 5.4, a contradiction. Now suppose qi is finite. We will prove the proposition for Si, which is indistinguishable from Shr. Define

V0 = 0 Tn = > Vn :’ Xt = i n = 0,1,..., Vn = inf(t > i n = 0,1, , ...

For simplicity we assume i is recurrent and qi ? 0 so that both and will be finite valued for all n. As is closed in Tn Vn S+i Orr , XT E S+i a.s. for all n, hence XT - i and Proposition 5.6 is applicable. Using this and Proposition 5.4 we see Vn - Tn is a sequence of independent exponential random variables with parameter qi. Thus Si is a union of intervals, closed on the left, and, as ~(Vn - Tn~ = oo, there can be at most finitely many in any finite time. These intervals must also be open on the right, for otherwise = i~ > 0 for some n, implying by the strong Markov property that remains in i for some time after a contradiction. X V, qed 310

REFERENCES

[1] Blumenthal, R.M. and Getoor, R.K. Markov Processes and Potential Theory, Academic Press, 1968. [2] Chung, K.L. Markov Chains with Stationary Transition Probabilities 2nd ed. Springer-Verlag, 1967.

[3] Chung, K.L. and Walsh, J.B. To reverse a Markov process, Acta Math. 123 (1969) 225-251. [4] Doob, J.L. Separability and measurable processes, J. Fac. Sci, U. of Tokyo (to appear.) [5] Meyer, P.A. Probability and Potentials, Ginn, Blaisdell 1966. [6] Stroock, D. The Kac approach to potential theory, J. Math. Mech. 16 (1967) 829-852.